most likely this is caused by a bent brake rotor on one or both sides. to fix this remove the tire and the brake calliper and the rotor will just slide off then just replace it and put everything back together.
maybe the car need to be linged up n check the power steering.
Your AC should have NOTHING to do with your barking system, as it is not tied into the vacuum of the engine. If your wheel shakes under braking, then you need to replace your rotors and pads.
All of the following causes mean you have a serious safety issue: Most comon is a bent brake rotor, but bent drums are also a suspect. Oil leaking onto the brakes can all cause this.
Warped rotors
During acceleration? I'm not sure. But when both idling and acceleration the engine's combustion can cause a lot of vibration in the steering wheel if you are in an older vehicle.
You may need a wheel alignment, or you may need bushings.
Could be motor mounts, the tires out of alignment or out of balance, or the steering needs work.
my dodge has the same problem, it is the power steering return hose being cracked or cut.
Power steering line leak, need to fix as can cause fire.
This would normally be down to either: Warped Brake Discs (in some counties these are called Rotors) where there is a small ripple in the surface of the disc. This causes the brake pads (which are flat) to make intermittant contact with the surface which causes a vibration. This "warping" is not always visible to the naked eye. Worn or Perrished suspension compnents. This can lead to the wheels moving around in an unctrolled manner during braking which can also cause vibration.
Vibration at idle, especially with automatic transmission, is usually caused by engine idling to slow. Now, since there is no idle adjustment screw anymore you need to have the vehicle checked at your Toyota Dealer. Some vibration, even to that level is normal. IF IT HAPPENS IN THE PROCESS OF STOPPING, WHILE BRAKING, YOUR BRAKE ROTORS ARE LIKELY OUT OF ROUND.
What are the causes and sources of vibration on diesel